Question: $f(x)=\dfrac{1}{1-2{{x}^{2}}}$ Find a power series for $f$. Choose 1 answer: Choose 1 answer: (Choice A) A $ 1-2x^2+4x^4+\ldots +{{(-2)^n x }^{2n}}+\ldots$ (Choice B) B $1-2x+4x^2+\ldots +{{(-2)^n {{x}} }^{n}}+\ldots$ (Choice C) C $ 1+2{{x}^{2}}+4x^4+\ldots +{{2^n {{x}} }^{2n}}+\ldots$ (Choice D) D $ 1-2{{x}^{2}}-4x^4+\ldots -{{2^n {{x}} }^{2n}}+\ldots$
Solution: This is a geometric series with first term $a\text{ }=\text{ }1$ and common ratio $r\text{ }=\text{ }{2{x}^{2}}$. Therefore, the series is as follows. $1+2{{x}^{2}}+4x^4+\ldots +2^n x^{2n}+\ldots $